Qualitative Modelling and Simulation of Genetic Regulatory Networks - - PowerPoint PPT Presentation

Qualitative Modelling and Simulation of Genetic Regulatory Networks in Bacteria

Hidde de Jong1 Delphine Ropers1 Johannes Geiselmann1,2

1INRIA Grenoble-Rhône-Alpes 2Laboratoire Adaptation Pathogénie des Microorganismes

CNRS UMR 5163 Université Joseph Fourier Email: Hidde.de-Jong@inrialpes.fr

2

IBIS at INRIA Grenoble – Rhône-Alpes

IBIS: systems biology group at INRIA and Joseph Fourier University/CNRS

• Analysis of bacterial regulatory networks
• Interdisciplinary approach using models and experiments
• Group composed of computer scientists, biologists, physicists, …

3

Overview

• 1. Genetic regulatory networks in bacteria
• 2. Qualitative simulation of genetic regulatory networks

using piecewise-linear models

• 3. Qualitative simulation of carbon starvation response in

Escherichia coli: model predictions and validation

• 4. Conclusions and perspectives

4

Escherichia coli stress responses

• E. coli is able to adapt to a variety of stresses in its environment

Model organism for understanding of decision-making processes in single-cell organisms

• E. coli is easy to manipulate in the laboratory

Model organism for understanding adaptation of pathogenic bacteria to their host

Nutritional stress Osmotic stress Heat shock Cold shock …

Storz and Hengge-Aronis (2000), Bacterial Stress Responses, ASM Press 2 µm

5

Response of E. coli to carbon starvation

Response of E. coli to carbon starvation conditions: transition from exponential phase to stationary phase

Changes in morphology, metabolism, gene expression, …

log (pop. size) time > 4 h

6

Carbon starvation response network

Response of E. coli to carbon source availability is controlled by large and complex genetic regulatory network Network senses carbon source availability and global regulators coordinate adaptive response of bacteria

Ropers et al. (2006), Biosystems, 84(2):124-52

7

Carbon starvation response network

Response of E. coli to carbon source availability is controlled by large and complex genetic regulatory network No global view of functioning of network available, despite abundant knowledge on network components

Mathematical modeling and computer analysis and simulation

Ropers et al. (2006), Biosystems, 84(2):124-52

8

Modeling of genetic regulatory network

Well-established theory for modeling of genetic regulatory networks using ordinary differential equation (ODE) models Practical problems encountered by modelers:

• Knowledge on molecular mechanisms rare
• Quantitative information on kinetic parameters and molecular

concentrations absent

• Large models

Even in the case of well-studied E. coli network!

de Jong (2002), J. Comput. Biol., 9(1): 69-105 Hasty et al. (2001), Nat. Rev. Genet., 2(4):268-279 Smolen et al. (2000), Bull. Math. Biol., 62(2):247-292 Szallassi et al. (2006), System Modeling in Cellular Biology, MIT Press

9

Qualitative modeling and simulation

Possible strategies to overcome problems

• Parameter estimation from experimental data
• Parameter sensitivity analysis
• Model simplifications

Intuition: essential properties of network dynamics robust against reasonable model simplifications Qualitative modeling and simulation of large and complex genetic regulatory networks using simplified models Relation with discrete, logical models of gene regulation

Thomas and d’Ari (1990), Biological Feedback, CRC Press de Jong, Gouzé et al. (2004), Bull. Math. Biol., 66(2):301-40 Kauffman (1993), The Origins of Order, Oxford University Press

10

PL differential equation models

Genetic networks modeled by class of differential equations using step functions to describe regulatory interactions

xa = κa s-(xa , θa2) s-(xb , θb ) – γa xa

.

xb = κb s-(xa , θa1) – γb xb

.

x : protein concentration κ , γ : rate constants θ : threshold concentration

x s-(x, ) θ 1

Differential equation models of regulatory networks are piecewise-linear (PL)

Glass and Kauffman (1973), J. Theor. Biol., 39(1):103-29

11

Analysis of local dynamics of PL models

Monotonic convergence towards focal state in every region between thresholds θa1 maxb θa2 θb maxa

Mathematical analysis of PL models

xa = κa s-(xa , θa2) s-(xb , θb ) – γa xa

.

xb = κb s-(xa , θa1) – γb xb

.

θa1 maxb θa2 θb maxa κa/γa κb/γb

xa = κa – γa xa

.

xb = κb – γb xb

.

D1

Glass and Kauffman (1973), J. Theor. Biol., 39(1):103-29

12

Analysis of local dynamics of PL models

Monotonic convergence towards focal state in every region between thresholds θa1 maxb θa2 θb maxa

Mathematical analysis of PL models

xa = κa s-(xa , θa2) s-(xb , θb ) – γa xa

.

xb = κb s-(xa , θa1) – γb xb

.

xa = κa – γa xa

.

xb = – γb xb

.

θa1 maxb θa2 θb maxa κa/γa

D5

Glass and Kauffman (1973), J. Theor. Biol., 39(1):103-29

13

Analysis of local dynamics of PL models Extension of PL differential equations to differential inclusions using Filippov approach

θa1 maxb θa2 θb maxa

Mathematical analysis of PL models

xa = κa s-(xa , θa2) s-(xb , θb ) – γa xa

.

xb = κb s-(xa , θa1) – γb xb

.

θa1 maxb θa2 θb maxa

D3

Gouzé and Sari (2002), Dyn. Syst., 17(4):299-316

14

Analysis of local dynamics of PL models Extension of PL differential equations to differential inclusions using Filippov approach

θa1 maxb θa2 θb maxa

Mathematical analysis of PL models

xa = κa s-(xa , θa2) s-(xb , θb ) – γa xa

.

xb = κb s-(xa , θa1) – γb xb

.

θa1 maxb θa2 θb maxa

D7

Gouzé and Sari (2002), Dyn. Syst., 17(4):299-316

15

State space can be partitioned into regions with unique derivative sign pattern Qualitative abstraction yields state transition graph that provides discrete picture of continuous dynamics

θa1 maxb θa2 θb maxa

Qualitative analysis of PL models

. .

xa > 0 xb < 0 D5:

. . . . . .

xa > 0 xb > 0 xa > 0 xb < 0 xa = 0 xb < 0 D1: D5: D7:

θa1 maxb θa2 θb maxa

D12 D22 D23 D24 D17 D18 D21 D20 D1 D3 D5 D7 D9 D15 D27 D26 D25 D11 D13 D14 D2 D4 D6 D8 D10 D16 D19

D1 D3 D5 D7 D9 D15 D27 D26 D25 D11 D12 D13 D14 D2 D4 D6 D8 D10 D16 D17 D18 D20 D19 D21 D22 D23 D24 de Jong et al. (2004), Bull. Math. Biol., 66(2):301-40 Batt et al. (2008), Automatica, 44(4):982-89

16

State transition graph gives conservative approximation of continuous dynamics

• Every solution of PL model corresponds to path in state transition graph
• Converse is not necessarily true!

State transition graph is invariant for given inequality constraints on parameters

Qualitative analysis of PL models

D1 D3 D11 D12

θa1 maxb θa2 θb maxa θa1 maxb θa2 θb maxa κa/γa κb/γb

D1 D11 D12 D3

0 < θa1 < θa2 < κa/γa < maxa 0 < θb < κb/γb < maxb

Batt et al. (2008), Automatica, 44(4):982-89

17

State transition graph gives conservative approximation of continuous dynamics

• Every solution of PL model corresponds to path in state transition graph
• Converse is not necessarily true!

State transition graph is invariant for given inequality constraints on parameters

Qualitative analysis of PL models

D1 D3 D11 D12

0 < θa1 < θa2 < κa/γa < maxa 0 < θb < κb/γb < maxb

θa1 maxb θa2 θb maxa θa1 maxb θa2 θb maxa κa/γa κb/γb

D1 D11 D12 D3

Batt et al. (2008), Automatica, 44(4):982-89

18

State transition graph gives conservative approximation of continuous dynamics

• Every solution of PL model corresponds to path in state transition graph
• Converse is not necessarily true!

State transition graph is invariant for given inequality constraints on parameters

Qualitative analysis of PL models

D1 D11

θa1 maxb θa2 θb maxa θa1 maxb θa2 θb maxa κa/γa κb/γb

D1 D11 D12 D3

0 < κa/γa < θa1 < θa2 < maxa 0 < θb < κb/γb < maxb

Batt et al. (2008), Automatica, 44(4):982-89

19 D16 D18 D20

Use of state transition graph

Analysis of steady states and limit cycles of PL models

• Attractor states in graph correspond (under certain conditions) to stable

steady states of PL model

• Attractor cycles in graph correspond (under certain conditions) to stable

limit cycles of PL model θa1 maxb θa2 θb maxa θa1 maxb θa2 θb maxa

D1 D3 D5 D7 D9 D15 D27 D26 D25 D11 D12 D13 D14 D2 D4 D6 D8 D10 D17 D19 D21 D22 D23 D24 Casey et al. (2006), J. Math Biol., 52(1):27-56 Glass and Pasternack (1978), J. Math Biol., 6(2):207-23 Edwards (2000), Physica D, 146(1-4):165-99

20 D1 D3 D5 D7 D9 D15 D27 D26 D25 D11 D12 D13 D14 D4 D6 D8 D10 D16 D17 D18 D20 D19 D21 D22 D23 D24

Use of state transition graph

Determine reachability of attractors of PL models from given initial states: qualitative simulation Simulation algorithms based on symbolic computation instead

• f numerical simulation

Use of inequality constraints between parameters

D2 Kuipers (1994), Qualitative Reasoning, MIT Press

21

Paths in state transition graph represent predicted sequences

• f qualitative events

Model validation: comparison of predicted and observed sequences of qualitative events Need for automated and efficient tools for model validation

D1 D3 D5 D7 D9 D15 D27 D26 D25 D11 D12 D13 D14 D2 D4 D6 D8 D10 D16 D17 D18 D20 D19 D21 D22 D23 D24

Use of state transition graph

. .

xa < 0 xb > 0 xa > 0 xb > 0 xa= 0 xb= 0

. . . .

D1: D17: D18:

Concistency?

Yes

xb

time time

xa xa > 0

.

xb > 0

.

xb > 0

.

xa < 0

.

22

Model validation by model checking

Dynamic properties of system can be expressed in temporal logic (CTL) Model checking is automated technique for verifying that state transition graph satisfies temporal-logic statements

Efficient computer tools available for model checking

There Exists a Future state where xa > 0 and xb > 0 and starting from that state, there Exists a Future state where xa < 0 and xb > 0

. . . .

EF(xa > 0 ∧ xb > 0 ∧ EF(xa < 0 ∧ xb > 0) )

. . . .

xb

time time

xa xa > 0

.

xb > 0

.

xb > 0

.

xa < 0

.

Batt et al. (2005), Bioinformatics, 21(supp. 1): i19-i28 Chabrier et al. (2004), Theor. Comput. Sci., 325(1): 25-44

23

Genetic Network Analyzer (GNA)

http://www-helix.inrialpes.fr/gna

Qualitative analysis of PL models implemented in Java: Genetic Network Analyzer (GNA)

de Jong et al. (2003), Bioinformatics, 19(3):336-44

Distribution by Genostar SA

24

Analysis of bacterial regulatory networks

Applications of qualitative simulation in bacteria:

• Initiation of sporulation in Bacillus subtilis
• Quorum sensing in Pseudomonas

aeruginosa

• Onset of virulence in Erwinia

chrysanthemi

de Jong, Geiselmann et al. (2004), Bull. Math. Biol., 66(2):261-300 Viretta and Fussenegger (2004), Biotechnol. Prog., 20(3):670-78 Sepulchre et al. (2007), J. Theor. Biol., 244(2):239-57

25

Modeling of carbon starvation network

Can we make sense of the carbon starvation response of E. coli using this simple network?

How do global regulators bring about growth arrest when carbon sources are running out?

Critical step: translation of network diagram into PL model

26

Model reduction strategies to obtain PL models Reduced model are easier to analyze, little loss of precision

Cya concentration (M) Crp concentration (M)

ODE model Reduced ODE model PL model

Quasi-steady-state approximation Piecewise-linear approximation Fast Slow

Modeling of carbon starvation network

12 variables, 46 parameters 7 variables, 43 parameters 7 variables, 36 parameter inequalities

27

Steady states of carbon starvation model

Analysis of steady states of PL model of carbon starvation network: two stable steady states

• Steady state corresponding to exponential-phase conditions
• Steady state corresponding to stationary-phase conditions

28

Transition to stationary phase

Does model reproduce transition from exponential phase to stationary phase upon carbon starvation?

log (pop. size) time carbon run-out

29

Simulation of carbon starvation network

Qualitative simulation of carbon starvation response

State transition graph with 27 states, starting from exponential phase, all paths converge to stationary-phase steady state upon stress signal

CYA FIS GyrAB Signal TopA rrn CRP

30

Insight into carbon starvation response

Role of the mutual inhibition of Fis and CRP•cAMP in the adjustment of growth of cells following carbon starvation

Sequence of qualitative events predicted by qualitative simulation

31

Validation of carbon starvation model

Validation of model using model checking

• “Fis concentration decreases and becomes steady in stationary phase”
• “cya transcription is negatively regulated by the complex cAMP-CRP”
• “DNA supercoiling decreases during transition to stationary phase”

EF(xfis < 0 ∧ EF(xfis = 0 ∧ xrrn < θrrn) )

. .

True AG(xcrp > θ3

crp ∧ xcya > θ3 cya ∧ xs > θs

EF xcya < 0)

.

True False EF( (xgyrAB < 0 ∨ xtopA > 0) ∧ xrrn < θrrn)

. .

Ali Azam et al. (1999), J. Bacteriol., 181(20):6361-6370 Kawamukai et al. (1985), J. Bacteriol., 164(2):872-877 Balke, Gralla (1987), J. Bacteriol., 169(10):4499-4506

32

Suggestion of missing interaction

Model does not reproduce observed downregulation of supercoiling

Missing interaction in the network?

33

Extension with new global regulator

Missing component in the network?

Model does not reproduce observed downregulation of supercoiling

34

Transition to exponential phase

Does model reproduce reentry into exponential phase after carbon upshift of stationary-phase cells?

log (pop. size) time carbon upshift

35

Simulation of carbon upshift

Qualitative simulation of carbon upshift

• State transition graph with hundreds of states: starting from stationary

phase and upshift conditions, all paths converge to cyclic attractor

• Cyclic attractor corresponds to damped oscillations towards exponential-

phase steady state

CYA FIS CRP GyrAB Signal TopA rrn

steady state steady state

36

Insight into carbon upshift

Role of negative feedback loops involving Fis, GyrAB and TopA in the adjustment of growth of cells after carbon upshift

Sequence of qualitative events predicted by qualitative simulation

37

Novel predictions

Predictions of damped oscillations cannot be verified with currently available experimental data

Idem for certain predictions on transition to stationary phase

CYA FIS CRP GyrAB Signal TopA rrn

steady state steady state

38

Perspectives: experimental verification

Real-time monitoring of gene expression using fluorescent and luminescent reporter gene systems

Global regulator GFP

• E. coli

genome Reporter gene

excitation emission

Collaboration with UJF

39

Perspectives: model checking

Make model-checking technology available to modelers of biological systems

• Integrate model checkers in modeling and simulation tools
• Develop temporal logics tailored to biological questions
• Develop temporal-logic

patterns patterns for frequently-asked modeling questions

Calzone et al. (2006), Bioinformatics, 22(14):1805-7 Monteiro et al. (2008), Bioinformatics, 24(16):i227-33 Mateescu et al. (2008), ATVA 2008, LNCS 5311, 48-63

40

Perspectives: network identification

Adaptation of methods for hybrid-systems identification to PL models of genetic regulatory networks

Time-series data

Switch detection Mode estimation Threshold reconstruction Network inference

Drulhe et al. (2008), IEEE Trans. Autom. Control, 53(1):153-65 Porreca et al. (2008), J. Comput. Biol., in press

41

Conclusions

Modeling of genetic regulatory networks in bacteria often hampered by lack of information on parameter values Use of coarse-grained PL models that provide reasonable approximation of dynamics Mathematical methods and computer tools for analysis of qualitative dynamics of PL models

Weak information on parameter values (inequality constraints)

Use of PL models may gain insight into functioning of large and complex networks PL models provide first idea of qualitative dynamics that may guide modeling by means of more accurate models

42

Contributors and sponsors

Grégory Batt, INRIA Rocquencourt Valentina Baldazzi, INRIA Grenoble-Rhône-Alpes Bruno Besson, INRIA Grenoble-Rhône-Alpes Hidde de Jong, INRIA Grenoble-Rhône-Alpes Estelle Dumas, INRIA Grenoble-Rhône-Alpes Johannes Geiselmann, Université Joseph Fourier, Grenoble Jean-Luc Gouzé, INRIA Sophia-Antipolis Radu Mateescu, INRIA Grenoble-Rhône-Alpes Pedro Monteiro, INRIA Grenoble-Rhône-Alpes/IST Lisbon Michel Page, INRIA Grenoble-Rhône-Alpes/Université Pierre Mendès France, Grenoble Corinne Pinel, Université Joseph Fourier, Grenoble Caroline Ranquet, Université Joseph Fourier, Grenoble Delphine Ropers, INRIA Grenoble-Rhône-Alpes Ministère de la Recherche, IMPBIO program European Commission, FP6, NEST program INRIA, ARC program Agence Nationale de la Recherche, BioSys program